forces on bubbles growing and detaching in flow along a ...it is found that either a vorticity lift...

Forces on bubbles growing and detaching in flow along avertical wallCitation for published version (APA):Helden, van, W. G. J., Geld, van der, C. W. M., & Boot, P. G. M. (1995). Forces on bubbles growing anddetaching in flow along a vertical wall. International Journal of Heat and Mass Transfer, 38(11), 2075-2088.https://doi.org/doi:10.1016/0017-9310(94)00319-Q, https://doi.org/10.1016/0017-9310(94)00319-Q

DOI:doi:10.1016/0017-9310(94)00319-Q10.1016/0017-9310(94)00319-Q

Document status and date:Published: 01/01/1995

Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne

Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.

Download date: 28. Aug. 2021

https://doi.org/doi:10.1016/0017-9310(94)00319-Q

https://doi.org/10.1016/0017-9310(94)00319-Q

https://doi.org/doi:10.1016/0017-9310(94)00319-Q

https://doi.org/10.1016/0017-9310(94)00319-Q

https://research.tue.nl/en/publications/afe853f8-9b76-4a6a-95cf-efb3dfe5308d

Pergamon hf. 1. Heat Mass TranrJk Vol. 38, No. 1 I, pp. 2075 -21188, 1995

Copyright Q 1995 Elsewr Science Ltd Printed in Great Britain. All rights reserved

@017-9310’955950+000

0017-9310(94)00319-x

Forces on bubbles growing and detaching in flow along a vertical wall

W. G. J. VAN HELDEN, C. W. M. VAN DER GELD7 and P. G. M. BOOT Eindhoven University of Technology, Faculty of Mechanical Engineering, W-hoog 3.144,

P.O. Box 513, 5600 MB Eindhoven, The Netherlands

(Received 22 June 1994 and injnalforrn 14 September 1994)

Abstract-Experiments are performed on bubble detachment from an artificial cavity in a plane wall of a vertical rectangular channel. Mean upward velocity is varied. Steam bubbles are generated by local heating of the cavity, nitrogen bubbles of about the same size by injection. The experiments show a difference in take off direction between vapor and nitrogen bubbles. Steam bubbles take off into the liquid, while nitrogen bubbles more or less slide parallel to the wall. The bubble detachment radius decreases for increasing bulk liquid velocity, in a way that merely depends on the detachment radius without convection. Nitrogen bubbles, coming from a capillary with approximately the same radius are larger than water vapor bubbles. A force coefficient fit is performed on force components perpendicular to the wall. By analyzing flow- and non-flow experiments separately, some of the forces are quantified. By combining the results of nitrogen bubble and steam bubble experiments, a force due to the temperature difference at the bubble foot is studied. Such a force could explain the observed differences between steam and nitrogen bubbles. It is found that either a vorticity lift force of the type found by Auton [The dynamics of bubbles, drops and particles in motion in liquids, Ph.D. thesis, University of Cambridge (1983)] is negligible, or this temperature difference force may be important. A commonly used criterion to predict detachment radii is

found not to be satisfactory.

1. INTROWCTION As boiling heat transfer occurs frequently in various industrial processes, fundamental knowledge of bubble growth and detachment from heated surfaces is important. Better knowledge may help to improve efficiency and reliability of boiling processes.

In many industrial applications boiling takes place in vertical pipes. However, most experiments on bubble detachment were performed on horizontal surfaces. Gravity then plays an important, if not domi- nant, role. Bubbles are nonetheless reported to detach against gravity from downward facing plates or from heated wires [Z] ; even in microgravity, bubbles detach as a result of self-induced convection [3]. Clearly, detachment without the action of gravity occurs, as is further discussed below.

Siegel [2] studied boiling heat transfer in gravity, reduced to as low as 0.001 g. At low g, bubbles were found to detach relatively slowly, resulting in coales- cence of subsequent bubbles. Growth rate and detachment diameter increased with decreasingg. Siegel still considered buoyancy as the main cause of bubble detachment.

Cooper and Chandratilleke [3] observed bubbles detaching from a horizontal, upward faced surface in n-hexane in a known temperature gradient. In their experiments they reduced the gravitational acceleration to about 0.4% of Earth gravity. In micro-

t Author to whom correspondence should be addressed.

gravity, bubble shape and departure were explained with the aid of the fully developed temperature boundary layer and wall superheating, without buoyancy. In microgravity this thermal boundary layer was found to have a dominating effect on growth and departure [3]. Detachment ‘against’ gravity was found to be stimulated by strong subcooling of the liquid and by a thin thermal boundary layer.

In the present study, bubbles detach at ambient pressure from an artificial cavity in a vertical plane wall under 1 g and the importance of gravity and other forces in these circumstances is investigated. Some of these forces are due to interfacial stresses created by liquid flowing upward along the wall.

For instance the effect of convection on bubble detachment was investigated by Koumoutsos et al. [4]. They studied bubbles detaching from a horizontal wall and found the detachment diameter to decrease with increasing liquid velocity. Al-Hayes and Win- terton [5] studied gas bubbles taking off into a flowing liquid. The inclination of their test tube was varied between -30 and +30’ from the horizontal. Their interpretation was based on the combined action of buoyancy, surface tension and drag forces. Detach- ment diameters were reproduced reasonably well. However, a maximum detachment diameter was predicted to occur with increasing downward flow, which was not observed experimentally.

Cooper et al. [6] performed a series of measurements on growing bubbles in supersaturated n- hexane, with and without gravity, in fully developed

2075

2076 W. G. J. VAN HELDEN et al.

A C c c, D E e F

g h H

Kl

k 1

T 4 Q R Re T u

v, v v V/*

NOh4ENcLATlJRE

area of cross section specific heat constant coefficient growth constant diameter constant unit vector force gravity acceleration enthalpy of evaporation height distance from bubble top to capillary axis heat transfer coefficient width of channel mass pressure heat flux heat flow rate radius Reynolds number temperature velocity in x-direction velocity in y-direction volume friction velocity.

Greek symbols

; thermal diffusivity contact angle

; growth exponent boundary layer thickness

0 dimensionless temperature V kinematic viscosity

4 exponent P mass density cr surface tension r shear stress 4 circumferential angle rD take-off angle w vorticity.

Subscripts a advancing, upstream b bubble bulk liquid bulk B buoyancy cap capillary m bubble midpoint

: net, total at detachment

D drag e expansion g gas-vapor H hydraulic 1 liquid L lift r receding, downstream rel relative sat at saturation top at bubble top W wall X,Y x-, y-direction cc at infinity, undisturbed.

Superscripts - mean.

laminar flow. At 1 g, bubbles were found to slide along the vertical wall, forming a microlayer at the bubble foot. The drag force on these bubbles was found to be larger than for unbounded bubbles. Viscous forces in the microlayer wedge were held responsible for this. From measurements under microgravity they concluded that bubble growth is merely dependent on the Jakob number [defined as p,c(T- T,t)/@&)] and liquid thermal diffusivity, with negligible influence of viscosity and surface tension.

More recently, Klausner et al. [7] studied the detachment of bubbles from a horizontal heated wall into saturated flowing R113. Their heater was a NiCr strip without artificial cavities. They found the majority of the bubbles first to move parallel, to the wall with their foot still at the wall before lifting off. Because no fixed nucleation site was studied, their results were stated in terms of probability density functions. An increase of wall heat flux or a decrease of liquid mass flux was found to cause a shift of the mean detachment diameter towards larger values.

From a force balance, this result was predicted with main parameters being the liquid velocity and the wall superheat. Of main importance in their force balance parallel to the wall was the force resulting from asymmetrical bubble growth. The same force balance was used by Zeng et al. [8] to predict bubble detachment diameters in the horizontal flow boiling of R113. In the present study, the importance of the asymmetrical bubble growth is re-evaluated.

From the foregoing it is concluded that in situations where buoyancy is relatively unimportant, the effect of wall superheating may be important in bubble detachment. The relative importance of induced temperature gradients is expected to be large in the case of bubbles detaching perpendicular to a vertical wall, since buoyancy in this case is mainly active parallel. In the present study the importance of forces resulting from temperature gradients is examined. To this end the actions of forced convection and heat transfer are disentangled by means of dedicated experiments.

Growing and detaching bubbles 2077

2. EXPERIMENTAL

Experiments are performed in a loop mainly com- prising stainless steel piping with 40 mm inner diameter. See the schematic of Fig; 1.

Water is pumped around and heated electrically to saturation temperature. The water then passes the test section with a square cross section of 20 x 20 mm2 (Fig. 2) and subsequently the condenser. The distance from round-square transition at the test section entrance to the bubble generation site is approximately 50 cm. For the liquid flow rates employed this is long enough to guarantee a fully developed flow along the bubbles. The channel Reynolds number in the experiments is varied between 11000 and 58 000. As the condenser is open to atmospheric pressure, the static pressure at the bubble generation site is roughly equal to the sum of atmospheric pressure and the

Condenser _ L-1

Heater Valve I 1

I+mtP ttttttt

Q Fig. 1. Schematics of the test loop with its main components.

Fig. 2. Schematic view of the test section.

water head above the bubble generator. The dissolved air is removed by boiling the water several hours prior to measurement. Then the flow is set to the desired value and the heater power is changed step by step until the liquid bulk temperature is stable and as near as possible to the saturation value. The bulk liquid superheat at the bubble generator position was always less than 2K.

Bubbles are generated in the test section (Fig. 2) with the aid of a special, flush-mounted bubble generator. Two parallel glass panels allow the filming of bubble growth and take-off. The inner sides of the test section walls are polished smoothly to minimise the number of unwanted nucleation sites. Surface rough- ness is of the order of 15 pm.

Two types of bubble generators are employed :

The bubble generator of type I (Fig. 3) consists of a capillary in a glass rod that is cast in a stainless steel plug. This plug is flush mounted in the test section wall and leaves a slit of less than 1.5 pm. The capillary mouth serves as an artificial nucleation site. Bubble generation is stimulated by heating of the capillary by means of a miniature coil inserted at the back of the capillary. The distance between electrical heating coil and bubble nucleation site is approximately 1.5 mm. The temperature of the heating coil is measured by a copper-constantane thermocouple in the centre of the coil (accuracy 2K). The type II bubble generator forms nitrogen bubbles by direct injection of nitrogen gas through a glass capillary. Identical plugs are used for type I and type II generators in order to create the same experimental conditions for nitrogen and steam bubbles.

The process of bubble growth and detachment is recorded by a high speed camera (Hitachi 16-DS) operated at a speed up to 5500 frames per second. Individual pictures are analyzed with the aid of a motion analyzer. Bubble contour and position coor-

mi Steel

\ ’ Plug

Fig. 3. Cross-section of type I bubble generator. The left side is the back of the generator, not in contact with the flow.


Table 1. Results for the vapor bubble experiments, arranged in sder of increasing liquid bulk speed. TEo,, is the heating coil temperature, Rd is t& mean detachment radius. The uncertainty value is the standard deviation. C,

and 7 are determined from R(t) = Ra + C,.t'

Experiment vbulk no. (mm s-‘)

N33 160 N35 180 N36 260 NO3 530 NO7 520 NO4 530 NO1 660 NO5 650 NO8 660 N27 700 N32 770 NO6 840 NO9 850 NO2 850

240 240 150 220 340 150 230 340

220 280 340

Rd W-4

c, (m s-l) p

0.558 kO.006 0.0054 1.0 0.504f0.007 o.oofJ4 1.5

0.40+0.01 0.00013 1.4 0.28 +O.Ol 0.052 1.1

0.282+_0.006 0.07 1.2 0.30+0.01 0.023 1.0

0.255 + 0.006 0.093 1.3 0.25 +0.02 0.066 1.4 0.25 + 0.04 0.0089 1.0 0.21 kO.03 0.04 1.2

0.2kO.02 0.042 1.3 0.19+0.01 0.54 1.4 0.18 +0.01 0.28 1.2 0.21 kO.02 0.04 1.0

dinates are fed into a computer and coupled to time as determined with the aid of timing dots produced by a stroboscope (accuracy 0.25 ms). The positioning accuracy of the hairlines of the analyzer in com- bination with the total magnification leads to a coordinate accuracy of 30 pm, approximately.

With vapor bubbles, a bubble generator with a capillary of 130 pm diameter is used. The water bulk speed is varied from 160 to 850 mm SK’ (accuracy 10 mm s-’ as measured by a rotameter), the heating coil temperature from 150 to 340°C. Nitrogen bubble experiments are performed in the same velocity range, with capillaries of 96, 101 and 132 pm diameter.

3. EXPERIMENTAL RESULTS

Table 1 lists the values found for the mean detachment radius in boiling. These are averages of 7-10 bubbles of a tine film recording of an experiment. Because the heating of the coil generator is indirect, only an estimate of the local temperature can be given. To determine the growth rate of individual bubbles, the attached bubble is assumed to have the shape of a

truncated sphere, as depicted in Fig. 9. Its radius as a function of time is written as :

R= R,,+C;t’ (1)

with RO a constant of the order of the capillary radius, t the time, C, the growth constant and y the growth exponent, the three coefficients R,,, C, and y are found by applying a least-squares fit to the measured bubble radius. The mean values of C, and y over a series of bubbles in one experiment are calculated and listed in Tables 1 and 2.

Figure 4 shows relative bubble midpoint positions after detachment for both types of generators. The x- and y-positions of the bubble midpoint are relative to the midpoint position at the moment of detachment. Steam bubbles are apparently launched into the liquid resulting in a detachment direction closer to the normal. Nitrogen bubbles, on the other hand, slide along the wall. This observation marks the difference between the heated and the isothermal cases. The trajectories of Fig. 5 are obtained by applying a least- squares polynomial fit to the trajectory data of a complete set of bubbles from one experiment. Clearly the

Table 2. Mean detachment radius found for nitrogen bubbles. Three series with different capillary radius are shown

Experiment R,, VW, Rd c no. pm (mm s-‘) (mm) (m SF) 1

N31 48 130 0.32+0.01 0.002 0.46 N30 48 700 0.251tO.01 0.015 0.63 N62 50 0 1.248 + 0.004 0.009 0.46 N60 50 570 0.55+0.02 0.039 0.66 N61 50 900 0.40 If: 0.02 0.083 0.75 N68 66 0 1.654+0.002 0.013 0.49 N65 66 540 0.7s+o.o2 0.049 0.69 N66 66 720 0.65 + 0.02 0.080 0.75 N67 66 980 0.52 + 0.02 0.213 0.88


h .z 5 o 2 I! J 0

B a 1 ‘Z c

0

0.0 0.2 0.4 0.6 0.8 1.0

Distance to wall minus Rd (mm)

Fig. 4. Trajectories of bubble midpoints after detachment, relative to the bubble midpoint at detachment. (A) water vapor bubbles from experiment NO1 (Table 1) ; (+) nitrogen bubbles from experiment N30 (Table 2). Note that the x-

and y-axis are differently scaled.

0.0 0.5 1.0 1.5

Dist. to wall minus R, (mm)

Fig. 5. Mean trajectories of relative bubble midpoint positions from four experiments. Solid line: nitrogen bubbles (N30). The other three lines are steam bubble trajectories : short dash, NO1 (Tcoi, = 150°C) ; long dash, NO5 (230°C) ;

dots, NO8 (340°C).

water vapor bubble take off direction is more towards receding angle. In case of nitrogen bubbles (Fig. 6), the normal than for nitrogen bubbles, while the take both pa and 8, decrease for increasing bubble radius. off angle (relative to the tangent to the wall) is slightly However, for steam bubbles B, is constant or slightly increased for increasing wall temperature. increasing, while fir decreases (Fig. 7).

For several individual bubbles, the advancing and receding contact angles are determined as a function of the bubble radius. These angles are defined in Fig. 9. Generally, the advancing contact angle exceeds the

The effect of liquid bulk velocity on bubble detachment radius is depicted in Fig. 8. The figure shows an inverse relation between detachment radius and liquid velocity. It is seen that, over the complete velocity

0.3 0.6 0.9 1.2 1.5

bubble radius (mm)

Fig. 6. B, (solid lines) and /3, (dashed lines), for nitrogen bubbles@,,= 1320).(+) VbulL=Oms-‘;(A)O.S4ms-‘;

(0) 0.72 m s-l; (V) 0.98 m s-l.

J

0.15 0.20 0.25

bubble radius (mm)

Fig. 7. /& (solid lines), and & (dashed lines), for two steam bubbles. (+) Vhulk = 0.65 m s-‘; (A) 0.7 m s-‘.

1.7 rV

0.1 i

0.0 0.2 0.4 0.6 0.8 1.0

v, Ws) Fig. 8. Mean bubble detachment radius plotted as function of liquid bulk speed. Water vapor: (+) Dsap = 130 m. Nitrogen: (0) Dmp = 101 pm; (0) Dcap = 132 pm. Values of water vapor bubbles are given in Table 1, those of nitrogen

bubbles in Table 2.


Table 3. Values for coefficients E, and E2 that describe the relation between Rd and VbUa

Set

N,, D,, = 101 pm N,, Dcap = 132 pm Steam, D,,, = 130 pm

Coefficient E, E2

1.24 f 0.04 1.3+0.1 1.64kO.05 1.28+0.08 0.67kO.03 1.6+0.1

Table 4. Angle of midpoint motion just before detachment (a) for the vapor bubble experiments (0” if normal to the wall and 90” if parallel to the

wall)

Experiment no.

NO3 NO7 NO4 NO1 NO5 NO8 NO6 NO9 NO2

V bulk (mm ss’) (z;; ;)

530 150 58+3 520 230 62+3 530 340 58+3 660 150 61k8 650 230 56+2 660 340 5052 840 220 6Ok3 850 280 52+1 850 340 58+2

range, vapor bubbles are smaller than nitrogen bubbles if they come from capillaries of comparable size. The detachment radius for a given capillary is written as an exponential function of bulk liquid speed :

Rd = El exp C-G * ~bulkh PI

E, denotes the detachment radius at zero liquid speed. Values of E, and E, are listed in Table 3.

As discussed above, vapor bubbles detach in a direction away from the wall. From the coordinates of the bubble midpoint at the first three pictures before detachment, the slope of the midpoint motion is determined by fitting a polynomial. The angle of detachment is defined to amount 0” if normal to the wall and 90” if parallel to the wall. The average angle of vapor bubbles of an experiment is tabulated in Table 4. Rather surprisingly, the angle is nearly constant for the whole range of experiments, i.e. independent of liquid velocity. Obviously the influence of increasing drag.force with increasing liquid velocity is dominated by some other mechanism. In the following it is attempted to establish the importance of this mechanism. Variation of the liquid bulk speed influences the hydrodynamic forces, but also the heat exchange rate from wall to fluid. This combined effect is accounted for in the analysis. It is noted that aspect ratios (ratio of bubble axis in y-direction to that in x- direction) between 0.8 and 1.25 are measured for both nitrogen and steam bubbles. Thus the trajectory differences cannot be explained by differences in shape.

4. THE FOfNFES ON A DETACHtW BW8LE

It is observed above that the detachment behaviour of steam bubbles differs from that of nitrogen bubbles. In this section all forces that can be important in the detachment process are discussed. Expressions and coefficients for the forces are introduced, which will be determined further in the next section. The coordinate axes are depicted in Fig. 9.

The lift force comprises all hydrodynamic forces acting in the direction transverse to the flow. It is partly generated by the suction on an adhering body of a uniform flow over the body (‘Bernoulli suction’), partly by the action of the vorticity in the approaching liquid flow :

C,, accounts for the ‘Bernoulli suction’, A is the area of the bubble cross section perpendicular to the flow, Rap is the capillary radius, R the bubble radius, p, is the fluid density, 7b the bubble volume, I’,,, the fluid velocity at the bubble midpoint, e, the unit vector in x-direction, CL, a constant and v,, the relative velocity of fluid with respect to the bubble.

C,, is computed using Bernoulli’s equation and potential flow theory. This simplification is allowed since the bubble Reynolds number is high (typically 400 for the lowest velocities), which also justifies the neglect of Saffman’s lift force [9]. The computation yields CL, = 1 l/8 for the initial hemispherical bubble, with A = $TR& (see Appendix A). If the bubble is growing and takes on the form of a truncated sphere, the value of this force is assumed to decrease inversely proportional to the bubble radius, in order to account for the liquid flow between the bubble and the wall. This is made explicit by the factor R,,,/R in equation

(3). The second lift force contribution can also be esti-

mated from potential flow theory. Auton [1] derived the value of CL, for a bubble in an unbounded flow with small shear rate :

CL, = 0.53. (4)

In the detachment of a bubble from a plane wall, the

Fig. 9. Geometry used in modeling the forces acting on the bubble. Water is flowing in the positive y-direction.


vorticity o can be approximated by the ratio of liquid velocity at the bubble cap and the bubble cap x-position (see Fig. 9) : w x v(X)/X. The relative velocity, u,i, is the water velocity at the bubble centre, V,.

The shear rate is defined as &,/Ybolr and should be small to justify the use of equations (3) and (4). In our situation, the assumption of a small shear rate would lead to the demand of Rb being smaller than the velocity turbulent boundary layer thickness 6, since w z Vbullr/8. From ref. [IO] this thickness is estimated for a channel Reynolds number of 4.2 x 104 to be x 5 mm in our setup. However, this is a global demand and especially near the wall the vorticity is much higher. Small bubbles will therefore experience a higher shear rate. This, combined with the fact that the flow in our situation is not unbounded, should cause a different value of CL,.

The surface tension force in two directions is given by:

-RcapcJ s

=cos/3cos#d&,,. (5) 0

Reap is the capillary radius, c the surface tension, b = B(b) the contact angle between the liquid-vapor interface and the wall and 4 is the circumferential angle along the capillary mouth (see Fig. 9).

The bubble shape is assumed to approximate a truncated sphere. From the experiments it was found that /l varied between 90” upstream and 43” downstream of the bubble. Values of /3 for intermediate 4 are calculated assuming that B is a linear function of 4. A higher order fit does not lead to more accurate results. The integrals in equation (5) are then evaluated analytically. A deviation from the linear dependence of /I on 4 is accounted for by an extra coefficient C, on the RHS of equation (5), that has to be of order 1.

The corrected buoyancy or volume force, F,,, is composed of gravity acting on the mass of the bubble,

and the corrected Archimedes force, &, on an adhering bubble that is calculated by integrating pressure over the entire bubble enveloping area, A,. This area comprises two parts : one over the water-vapor interface, Ai, and one over the capillary mouth, A,,. This yields :

in which p, is the static pressure in the liquid (a function of y), p,, the static liquid pressure at the bubble

top, pp the vapor pressure inside the bubble and g Earth’s gravitational acceleration. H,, is the vertical distance from capillary to bubble downstream side (bubble top, see Fig. 9). R,,, is the curvature radius of the bubble top part.

After rearrangement and application of Gauss’ integral theorem on the integral over Ai it is found that

All these pressure forces are the result of the inner product of static pressure with the outward directed normal vector. Summation of equations (6) and (8) gives for the volume force :

Fa = (P, -pdg& + -$ Ampex top

Expansion force A growing bubble experiences a dynamic pressure

difference over its surface. For an expanding vapor bubble in unbounded liquid this pressure difference is found by applying mass and momentum conservation, giving the well-known Rayleigh equation :

Apdyn a p,(~l?2+ RR)). (10)

If the radius R is known as a function of time the resulting expansion force on the bubble is found by integrating equation (10) over the bubble surface. Witze et al. [l l] did so for a spherical bubble touching a plane surface. They assumed a square-root growth model for the bubble :

R(t) = Cpt”2. (11)

The resulting adhering force was found to be

Fe = 0.29nC& (12)

In a model to predict bubble detachment from a horizontal wall Klausner et al. [7] introduce a constant, C,, in the expansion force :

r;, = np,R2(;C,d2+Rl?-j. (13)

C, accounts for the non-sphericity of the bubble and for the presence of the wall. From a fit with their experimental data they found a value for C, of 20/3 which is rather large, in view of equation (10). We also adopt equation (13), but shall determine C, differently. For the growth rate of individual bubbles, a power function with unknown exponent is assumed :

R(t) = R, + C, * P. (14)

From the experiments the values of CY and y of individual bubbles are determined by a least-squares procedure to fit the measured R(t). The values presented in Tables I and 2 are values averaged over a number of bubbles per experiment. In the next section, a coefficient fit is done with the individual values of


C, and y (note that R, is usually small compared to &). To this end, the expansion force is decomposed with the aid of the measured angle @ :

F, = F,,x cos (@)e, + Fes sin (@)e,. (15)

cb is the direction of the bubble midpoint displacement just before detachment. Table 4 lists the mean values of @ found for some of the steam bubble experiments. The added mass of a bubble growing at a wall is accounted for by the expansion force equation (13).

The drag force on a bubble attached to a wall is, by definition, in the direction of the liquid flow and is written as :

FD = C,fp,AV;. (16)

CD is a function of turbulence intensity, bubble Reyn- olds number and bubble shape. Al-Hayes and Win- terton [5] derived a drag coefficient for bubbles attached to the inner side of perspex tubes. For 20 < Re, < 400 they found Co x 1.22. The bubble Reynolds number in our situation is typically much higher, while flow separation is expected to cause a change in drag coefficient. However, in the next section only forces in the x-direction are considered and equation (16) therefore needs no further evaluation. Interfacial stresses exerted in the x-direction are accounted for by the lift force.

Temperature drop force In the fluid flowing along the wall a temperature

gradient exists due to heat losses through the test section walls. Near the artificial cavity the heating element induces other temperature gradients. The surface tension of pure liquids decreases with increasing temperature. A bubble attached to this cavity has surface tension varying at its interface if the temperature gradients? are not reduced by evaporation or condensation at the interface. and the vapor temperature may vary. In pure vapor bubbles with a (nearly) uniform vapor pressure the interface temperature can only vary close to the bubble foot, where thermodynamic nonequilibrium might cause the saturation equation, relating pressure and temperature, not to be valid anymore. The nitrogen bubbles have been injected at isothermal conditions. In this case, there is no surface tension variation nor any other effect due to temperature drop or heat transfer, e.g. through turbulence.

Varying surface tension induces flow in the fluid along the bubble, called thermocapillary or Mar- angoni flow. Various effects of this flow on bubble growth and detachment have been reported [12, 131. In the present study it is not claimed that thermocapillarity is important in bubble detachment,

t It is noted that a temperature gradient is also present at the wall surface in contact with the liquid near the bubble foot. Since solid-fluid interfacial tension is of the or&r u cos fi this temperature gradient also create3 tangential stresses, though at places where liquid velocity is zero.

although the momentum flux of the liquid jet away from the wall, induued by it, might play a role. The significance of temperature differences near a bubble is merely investigated by attributing a force to it and by trying to find out if it may help to interpret results. In the following, this force is taken to be linear to a typical temperature drop (just as in some Marangoni flows), in order to be able to quantify it. If the temperature drop force would be mainly due to momentum incoming at the bubble base, p,$, caused by evaporation at heat flux q = p,u,h, and if q would be proportional to the temperature drop, AT, then the force should be proportional to ATZ rather than AT. However, this ‘recoil’ force is estimated to be six orders of magnitude smaller than other forces.

The sole conclusion to be drawn with the following analysis is whether the temperature drop force is important or not.

An expression for the temperature drop force is derived in the following way. The bubble foot is a circle with radius &+,. Let AT denote the mean temperature difference between wall and fluid at the bubble foot and let the surface tension vary linearly with temperature. If the temperature drop force, FAT, would be related to thermocapillarity, it can be estimated by :

FAT = 2n&, FT CATATe, (17)

in which C,, is a coefficient of order 1 and au/aT the surface tension temperature gradient. Note that ATis a function of liquid velocity and heating rate Q. In Appendix B the following estimate for this function is derived :

AT = 0.6. Q - Vfulk (18)

with [ = -0.568. The coefficient C,, and exponent 5 will be considered further in the next section.

After initiation and initial rapid growth a bubble expands relatively slowly. Typical growth times for vapor bubbles are in the order of 5 ms. For atmospheric pool boiling in water with 10K superheating, Slooten [ 141 found the transition from inertial to convective bubble growth to occur at 6 ps after bubble initiation. Our observations yield transition periods of less than 0.3 ms, whereas actual bubble growth always exceeds 1 ms. Since in our study attention is focussed on bubble detachment, the inertial stage and added-mass forces are neglected, except for the fi- term in equation (13).

The expansion of the bubble induces an acceleration of the fluid. In flow conditions, the interaction of acceleration and vorticity generation at the bubble surface gives rise to the so-called Basset history force. Here this type of force is negligible since both acceleration and vorticity generation are small.

The effect of turbulence is manifesting itself on the time scale of the most energetic eddies in the wall region. The friction velocity, I/*, in our experiments

Growing and detaching bubbles ?083

was of the order of 0.03 m s-‘. V* is defined as &X, with T,,. the wall shear stress. Following Klausner et ai. [7j, the most energetic eddies are assumed to have a width 1 of about lOOv/V*, with v the kinematic viscosity of the liquid. The time scale for these eddies is ztUrb % I/V* s lOOv/(V*)* x 3 x lo-*s. Bubble growth time is of the order of 5 ms, smaller than rturb. Turbulence is therefore neglected. In the next section it is attempted to quantify the unknown coefficients in the forces with the aid of the experimental results of the previous section.

5. ASSeSSMENT OF FORCE COEFFfCtENl’S FROM MEASl#RWEWS

In this section, the force balance in the x-direction is constructed with the aid of expressions and coefficients introduced in the previous section. The aim of this study is to quantify coefficients that are connected to the different forces. These coefficients are fitted to our experimental data. A force connected to the temperature difference between wall and fluid is included in the force balance. It is tested whether this force can elucidate the difference in detachment behavior between nitrogen and steam bubbles. Fur- thermore, it is investigated in Section 5.2 whether a force balance can be used to calculated the bubble detachment radius.

A computer program determines the coefficients of the different forces and calculates statistical quantities of the regression procedure.

5. I. Analysis and results 5.1.1. Approach. In this section the force balance on

a growing bubble in x-direction is studied. The bubble is attached to a capillary and it is assumed that at every moment the force balance &F,& = d(mo,)/dt holds. The x-components are selected since bubble take-off is in the x-direction (see Fig. 4). The force balance perpendicular to the wall is therefore considered to be more important. Gravity, in addition, dominates in the y-direction, which all but facilitates the determination of the other forces. In addition, if 8, = p, (no deviation from a truncated sphere shape, see Fig. 9), immediate detachment occurs according to the balance in the y-direction. Accurate measurement of j?. and fir would therefore be important in considering the balance, but experimental inaccuracy has been shown to be significant.

The balance in the x-direction reads :

+F,,x(O+GAO = y (19)

of the forces of which only the order of magnitude is known. If the expressions for the different forces are copied from Section 4, the force balance gets the following form :

+2nR,,,$,,0.6*(? Vi,,,

+ @g-P,)No+ g Acap. 1 fOP I

(20)

Note that the added mass is accounted for by the second-last term in equation (20), the expansion force F,. The direction of the coordinate axes is depicted in Fig. 9. In the acceleration term above, d(mv,)/dt, v, is the bubble center of mass velocity and m the bubble mass. For this term the following estimate is made :

x p,(4nvR2d+4/3xR3 2 1

z 4/3np,R2(3&*+Ri?) x 4/3;F. (21)

with C, in equation (20) of order 1. The factor pI/pI demonstrates that the acceleration force is three orders of magnitude less than Fe. The acceleration force contribution, being less than the error in C,, is therefore neglected in the following.

The force balance in equation (20) is rewritten in terms of the coe&ients to be fitted and the remaining parts :

+ CA&dL 4 + un> (22)

where X, contains the terms without a coetficient and n numbers the data points. For each bubble measured, certain instants of time before detachment are selected and each time yields a data point, since equation (20) should hold at all times. X, contains the following quantities measured :

A’, = np,R3#cos(@)+ (pg-p,)gH,+ Y$ 1

A,. top I

(23)

To circumvent difficulties in the determination of bubble shapes and related quantities, such as & a restriction on the set of possible bubble shapes is made. In the present model the bubble is assumed to have the shape of a truncated sphere (see Fig. 9). The asymmetry of the bubble is only accounted for by including the measured upstream and downstream contact angles /$,and Br, respectively. The contact angles at intermediate circumferential angle 4 are lin-


Table 5. Bubble sets used in the coefficient determination

Number of Set no. bubbles Type Vbvlt

1 6 N2 =o 101 2 2 NZ =o 132 3 IO N2 #O 101 4 25 N2 #O 132 5 30 steam #O 130

Table 6. Results for the force balance minimization procedure on data sets 1 and 2. Nitrogen bubbles with

zero bulk liquid speed

Data set

Fixed coefficients

Resulting values

1% @N)

1 c,, =o C, = 0.6kO.l 0.49 CL, = 0 C, = 0.6+0.2 CAT= 0

2 C, = 0.6kO.l 0.49 C, = 0.6kO.4

early interpolated from pa to Br. The bubble growth rate coefficient C, and the growth rate exponent y [see equation (14)J are determined for all bubbles indi- vidually by fitting, and used to compute li and #at the instants of time selected. In this way, the uncertainty in individual measurements is suppressed. A similar procedure is applied to 8, and /3,.

The force balqnee equation (19) depends on time and on the coefficients C,, CLi, CLZ, C,, and C,. Esti- mates for these coefficients are found by a procedure based on strict application of the total force balance equation (19). At every instant of time the force balance for every bubble from each capillary is made to be nearly satisfied by an appropriate selection of the coefficients C. The way this is done is explained in the next section.

51.2. Procedure and results. The results from experiments on steam and nitrogen gas bubbles are divided into different data sets, in order to increase the accuracy by reducing the number of coefficients to be fit. Table 5 lists the sets. The division is made according to type of bubble (nitrogen or steam), zero or non- zero liquid bulk velocity and capillary radius. The coefficients, fitted to the data of each set, are presented in Tables 6-8. The tables list the data set(s) used, the value of coefficients held fixed and the results for the unknown coefficients. The residuals obtained by com- puting the right-hand-side of equation (22) with the fitted coefficients are averaged over all data of a data set to yield IEl.

The first two data sets comprise the nitrogen exper-

t The correlation coefficient between XL, and X,, is 0.01.

iments with zero bulk liquid speed (eight bubbles). With no liquid flowing past the bubble and no additional heating of the wall only buoyancy, the expansion force and surface tension act on the bubble. The correlation coefficient between the input data for the combined data sets is very high (0.99). The scatter plot in Fig. 10 illustrates why the correlation is high. In this figure the X,-values of the data points are plotted vs their X,-values. The data of set 2 are in the lower left corner, while the other data are grouped in the opposite corner. This grouping leads to a high correlation coefficient. Keeping the two data sets apart leads to lower correlation coefficients. This yields the average values C, = 0.6 If 0.1 and C, = 0.6 f0.2, see Table 6. Both coefficients are of order 1, indicating that the minimization procedure yields realistic values. The error indicated is the estimate for the standard deviation, given by the square root of the diagonal elements of the error matrix.

Figure 11 depicts the residual force on the bubbles from data sets 1 and 2, by using the C, and C, values of Table 6. This force decreases monotonely as a function of time for each bubble (each line segment in Fig. 11). This is due to the fact that parameters, such as C,, are constant in time, as explained above.

Next, experiments with nitrogen and non-zero liquid speeds are used to determine the lift coefficients CL, and CL2, see Table 7. The value of the coefficient for the ‘vorticity’ lift force, CL,, tends to the lowest value possible (set to lo-‘j), while its deviation is sub- stantially higher. Hence, it is allowed statistically to take CL1 equal to zero. The functions XL, and XL, of equation (22), which are connected to the lift coefficients, have a rather large correlation coefficient of 0.89. In Fig. 12, the scatter plot for the XL, values vs XL, for sets 3 and 4 is given. With sets 1 and 2 better results have been obtained above by splitting into two separate sets. This does not work for sets 3 and 4. That XL, and XL, are correlated implies that it is better to determine only C,, while keeping CL, fixed, or vice- versa. If the theoretical estimate of 1 l/S for CL, is fixed, the value of 0.1 for CL, is found. All results are summarized in Table 7. The two cases, CL2 = 0 and CL, = 11 /S are considered to be limiting cases and are both analyzed in the following.

Set number 5, containing data of 30 boiling bubbles, is used to examine the importance of the force connected to the local heating of the wall by the bubble generator (see Table 8). This force might be helpful in explaining the different angle of detachment and other differences between nitrogen and vapor bubbles, as discussed above. In two runs listed in Table 8, the exponent 5 is set to -0.6, which is the value resulting from the model of Appendix B. It is noted that Car should be of order 1. If CLI is set to zero, a very small value of C,, results, with a large uncertainty. If CL, is set to 1 l/S, C,, is found to be 0.9. This is a result of the expected order of magnitude. The low value of CA7 for CL, = 0 might be due to the rather high correlation (0.57) between the XL, and X,, parameters.?


Table 7. Values for fixed coefficients and estimated coefficients for nitrogen bubbles with non-zero bulk liquid speed, data sets 3 and 4

Data set

Fixed coefficients

Resulting values

c,, = l.VkO.2 7.0 3 and 4 C. = 0.6

C, = 0.6 CL, = 0.0 CL, = 10-5~o.05

CL, 11/8 CL, = 1.83_+0.05 7.0

= CL, = 0.1+0.01 7.5

Table 8. Results for coefficient fit with steam bubbles

Data set

5

Fixed Resulting 1% coefficients values (PN)

CL, = 1.83 e = -0.6 C,, = 6 x 10+&0.4 1.4 C, = 0.6 CL* = 0.0 ~=4~10~,c*,=5x10-~+14 1.39 C, = 0.6 CL, = 11/S 4 = -0.6 Cn, = 0.9+_0.3 1.2

CL, = 0.1 5 = -5, c,, =-0.2 +0.2 0.8 -

I:lj/+<++* 1 -40 -30 -u) -10

x, WIN)

Fig. 10. X, vs X,-values of the data points from sets 1 and 2.

1.0

E 3 0.5

f ra 0.0

-05 .

1 40

data point number

Fig. 11 Total force on the bubbles of sets 1 and 2. Values computed with C, = 0.6 and CL = 0.6.

0 10 20 30

XL, W’JN)

Fia. 12. X, - vs A’, -values of data in sets 3 and 4.

Data set 5 has also been analyzed by varying both C,, and the exponent c in FAT, making thefitting model non-linear. Although the residuals IF”1 are smaller than with 6 fixed, the uncertainty of C&r at the lower CL2-value is only increased, whereas the [- value of the higher CLI-value (0.1) is the preset mini- mum, i.e. - 5. This makes it clear that the data do not favor the determination of 5.

5.1.3. Discussion. From the nitrogen bubble experiments with no bulk liquid speed it is found that C, and C, have a value of 0.6 + 0.1 and 0.6 + 0.2, respectively. A better accuracy can only be achieved by more data and by improved knowledge of the local contact angles & and j&.

The description of both lift forces FL, and FL2 leads to data sets that are more or less correlated. Attribut- ing a lower value to FL, leads to a higher value of FL2 and vice versa. By considering realistic extremes, CL1 is found to be 1.8 + 0.2 approximately, and CL2 to be 0.1 or less. These values do not exclude the possibility that a lift force of the type FL2, see equation (3) is not active at all, so it is recommended to perform dedicated experiments in which the lift force is isolated, in order to increase accuracy.

According to the estimates of Appendix B, the action of the temperature drop force would result in 5 having a value of -0.6, approximately. With this value of l, and with CL, fixed to 1 l/8, the coefficient found for this force is CAT = 0.9kO.3, implying that an extra detaching force should be present in the case of water vapor bubbles. The results are not conclusive, however, since there is a dependency on CL2 and the value of CL, is not yet quantified accurately enough. It is clear that either FL, is not active at all: i.e. CL1 = 0 and FL, is relatively large, or C,, has a significant value. It can not be decided on basis of the present data which option corresponds to reality, but both options are equally interesting. The authors have a tendency to fancv the ‘C,, significant’ oution.


1;; 0.15 0.20 0.25

bubble radius (mm)

Fig. 13. The computed individual forces on a vapor bubble from set 5. Values for coefficients listed in third tine of Table

g arc used. (1) Fax; (2) F,,; (3) FAT; (4) FL*; (5) F,,,.

because of the correlation coefficient (see above) and the observed differences between detaching vapor and nitrogen bubbles.

With CL, = 0.1 and CAr = 0.9 the values of the individual forces as a function of the bubble radius are given in Fig. 13. Especially for small bubble radii, both lift forces, FL, and FL, are of minor importance. The vorticity lift force, FL2, and the temperature drop force, FAT, are of minor importance during the whole of the bubble growth. The expansion force, F,, is even smaller and is therefore not depicted. The low relative importances of FL, and F,, explain why it is difficult to establish their absolute value on the basis of experiments with detaching bubbles.

5.2. Prediction of detachment radius Crucial in predicting the detachment radius is the

detaching, criterion. Mostly this criterion was formulated [4, 5, 7, 81 in terms of the net force on the bubble : as soon as the net force is pointing away from the wall the bubble would take off. However, if all forces are correctly included in a force balance, the sum of these forces should be zero at all times. Note that the method of analysis of the previous section is based on this observation.

Let us now try to devise a practical detachment criterion in much the same way it was done by others before. A typical value of the standard deviation of the residual force F, is 2 pN. The criterion for bubble detachment is tentatively formulated as follows: a bubble is supposed to detach if F,, as given by the RHS of equation (22), exceeds 2 pN. The force balance equation (20) is based on some assumptions, one of them being that the bubble shape is a truncated sphere. If the residual force, F”, exceeds a threshold, this assumption is, according to the criterion, not valid anymore and detachment occurs.

This criterion has been applied to the 30 steam bubbles from the fifth set, with the result that 29 of the 30 bubbles were found not to detach, since the residual force always exceeded the threshold value or remained negative. After increasing the threshold value to 10 pN, seven bubbles did detach. The differences between calculated and measured values were, however, very large and nearly all were outside the

+_ 50% range. It is therefore concluded that prediction of bubble detachment diameters with a criterion like the one above is unsatisfactory, despite the fact that this was quite often done in the past 14, 5, 7, 81. The geometrical criterion of Chesters [ 151 would do better, since it is based on the (im)possibility of fitting interface shapes to the cavity mouth. This criterion, however, does not explain the actual forces involved.

What seems to be missing is knowledge of the forces that actually cause a liquid bridging over the cavity mouth, i.e. that cause detachment of a bubble. If the physics involved were fully understood, an appropriate detachment criterion could be devised.

6. CONCLlJSiONS Forces acting on bubbles attached to a capillary are

to some degree of accuracy quantified with the aid of a fitting procedure and a step-by-step approach with results of three types of experiments. Two types of experiments have been with nitrogen injection from capillaries of 100 and 132 pm diameter. The other experiments have been with boiling bubbles, generated by local heating of a 130 ,nm diameter capillary.

The surface tension force coefficient C,, as defined in equation (20), is found to satisfy 0.4 f C, < 0.8. These are considered to be realistic values ; the spread is attributed mainly to uncertainties in the values of the contact angles /Ia and fir. The expansion force coefficient, C,, is found to be 0.6 +O.l. This is less than the value found by Klausner et al. [7], but is considered to be more realistic since their value is much larger than the value for a bubble, expanding freely far from a wall. Two lift forces are analyzed : one a contribution of uniform flow described by potential theory, FL,, and the other due to the vorticity, FL,. The data sets for the two lift forces are found to be more or less correlated. Either CL, or CL, has therefore been fixed in the fitting procedure, to the theoretical value 1 l/8 and to 0, respectively. The option CL, = 0 is selected on basis of a fit without fixed coefficients ; this yields CL1 = 1.83 in the fitting procedure, which is close to 11/S. The option C,, = 11/S yields CL, = 0.1, which is still small. These findings suggest that the relation for a vorticity lift force, as described by Auton [l], may not be applicable to detaching bubbles with flow along a wall. Dedicated measurements of lift forces are required to settle this question.

The detachment radii of nitrogen bubbles and steam bubbles are found to be different under similar circumstances. Also the angle of the trajectory of a detaching bubble with the vertical wall is different for nitrogen and steam bubbles. These differences may be due to the action of a force connected to the local heating of the wall. An expression for this force is given by equation (17). For the temperature drop AT in this equation, a numerical analysis has yielded equation (18). Coefficient fits yield that either FL, = 0 or the temperature drop force is significant, i.e. C,, = 0.9. If C,> = 0 the temperature drop force


probably is of minor importance. To reach fully conclusive results, knowledge of the lift forces would have to be improved. As yet, the authors believe that the option CL, = 1 l/8, CL, = 0.1 and C,, = 0.9 is more realistic in view of the discrepancies observed between nitrogen and steam bubbles and for other reasons explained above.

It is shown that if a force balance like equation (20) is used to predict detachment radii, the prediction is unsatisfactory. The authors believe that such a prediction criterion lacks some of the physics involved. Its use is not recommended, despite its manifold occurrence in literature [4, 5, 7, 81.

REFEBENCES

1. T. R. Auton, The dynamics of bubbles, drops and particles in motion in liquids. Ph.D. Thesis, University of Cambridge (1983).

2. R. Siegel, Effects of reduced gravity on heat transfer. In Advances in Heat Transfer (Edited by S. P. Hartnett), Vol. 4. Academic Press, London (1967).

3. M. G. Cooper and T. T. Chandratilleke, Growth of diffusion-controlled vapour bubbles at a wall in a known temperature gradient, Znt. J. Heat Mass Transfer 24(9), 147.5-1492 (1981).

4. N. Koumoutsos, R. Moissis and A. Spiridonos, A study af bubble departure in forced-convection boiling, J. Heat

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

Transfer Trans. ASME, 223-230 (1968). R. A. M. Al-Hayes and R. H. S. Winterton, Bubble diameter on detachment in flowing liquids, Int. J. Heat Mass Transfer 24,223-230 (1981). M. G. Cooper, K. Mori and C. R. Stone, Behaviour of vapour bubbles growing at a wall with forced convection, ht. J. Heat Muss Transfer 26(10), 1489-1507 (1983). J. F. Klausner, R. Mei, D. M. Bernard and L. Z. Zeng, Vapor bubble departure in forced convection boiling, ht. J. Heat Mass Transfer 36(3), 651-661 (1993). L. Z. Zeng, J. F. Klausner, D. M. Bernhard and R. Mei, A unified model for the prediction of bubble detachment diameters in boiling systems--II, ffow boiling, ht. J. Heat Mass Transfer 36(9), 2271-2279 (1993). P. G. Saffman, The lift on a small sphere in a slow shear flow, J. Fluid Mech. 22(2), 385-400 (1965). A. Melling and J. H. Whitelaw, Turbulent flow in a rectangular duct, J. Fluid Mech. 78(2), 289-315 (1976). C. P. Witze, V. E. Schrock and P. L. Chambre, Flow about a growing sphere in contact with a plane surface, ht. .I. Heat Mass Transfer El 1637-1652 (1968). E. S. Gaddis, The effect of liquid motion induced by phase change and thermccapillarity on the thermal equi- librium of a vapour bubble, Int. J. Heat Mass Transfer 19 2241-2250 (1972). Y. S. Kao and D. B. R. Kenning, Thermocapillary flow near a hemispherical bubble, J. Fluid Mech. 53(4), 715- 735 (1972). P. C. Slooten, Departure of vapor and gas bubbles in a wide pressure range. Ph.D. Thesis, Eindhoven University of Technology (1984). A. K. Chesters, An analytical solution for the profile and volume of a small drop or bubble symmetrical about a vertical axis, J. Fluid Mech. 81(4), 609-624 (1977). Richard W. Miller, Flow Measurement Handbook (2nd Edn). McGraw-Hill, New York (1989).

APPENDIX A. UFT COEFFICIENT ON A HEdlSPHBRE IN POTENTIAL FLOW

The ‘Bernoulli suction’ lift coefficient C,,, as introduced As the water is heated around the cavity, liquid tem- in,equation (3) is derived in the following way. Suppose a perature gradients occur near the foot of the bubble. If water

Fig. Al. Spherical polar coordinate system used to calculate lift force on hemisphere.

potential flow along a sphere as depicted in Fig. Al. Far away from the sphere the Bow is pa&e1 in +x-direction, in the vicinity it is described by the flow around a dipole in the origin. The radius of the sphere is unity. The relations between Cartesian and spherical coordinates are :

x = r sin (8) cos (4)

)’ = r sin (0) sin (4)

z = rcos(6). (Al)

If U, is the undisturbed fluid speed in the x-direction, the stream function, Y, is given by :

. (‘42)

On the surface of the sphere the radial velocity is zero, while the tangential velocity is given by

uR=~lJmsin(@. (A3)

The lift force on the hemisphere with y 2 0 is found by integrating the pressure at the surface over the whole area. The pressure is found by application of Bernoulli’s law :

p--pm = fp, {ui - V’,} = ip, C)‘,{$sin’ (6)- 1). (A4)

By definition,

As A is the area of the body cross section perpendicular to the flow direction, A = $t. lBe lift force acts in the y-direction, therefore the inner product of the normal vector at the hemisphere surface and the unit y-vector has to be incor- porated in the integration, giving :

sin’ (0){$in’ (6) - 1) d@

APPENDIX B. BSTSMWTE FOR THE OFTHETEMFERAtllllEWIOQFOfKZONLfQlJlD

SPEED

W. G. J. VAN HELDEN et al.

Fig. Bl. Temperature boundary layer generated by local heating of the wall. The heated area is indicated by the dotted line in the wall. Left: fully developed. Middle: decrease of boundary layer thickness by heat consumption of growing bubble. Right: reduced thickness after bubble detachment.

L

is flowing past the bubble the temperature at the bubble foot will vary slightly along the bubble foot. In order to appreciate the importance of the temperature drop force, the water superheating at the foot, AT, has to be modeled as a function of bulk liquid speed, Vbulk, and heating rate Q. A twodimen- sional geometry is assumed. This approximation should hold near the upstream end of the bubble. At other places, fluid motion is also directed towards the bubble foot, which to some extent validates the two-dimensional assumption.

The superheating AT at the bubble foot depends on time. Before initiation of bubble growth, heat is accumulated in the wall and also in the liquid in which a temperature boundary layer develops (see Fig. Bl). In the following calculation, the temperature boundary layer is assumed to be fully developed and stationary (left drawing). In the other draw- ings, the effect of a bubble on this layer is depicted. The bubble takes up heat from the wall, giving the boundary layer less time to develop. The thinner layer results in a larger temperature drop along the bubble surface. For larger bubbles this temperature drop is less since detachment fre- quency is lower and there is more time for the layer to develop. This leads to increasing temperature drop with increasing liquid speed (and decreasing detachment radii).

The temperature distribution in the boundary layer is governed by the stationary convection-diffusion equation :

where a is the thermal diffusivity, different for wall and fluid, p. the mass density of the (solid) wall, c the heat capacity of the wall and 0 the dimensionless temperature defined by

0 = (T- T,)/T,

with T, the upstream (saturation) temperature. Q is the heat production rate of the source (in J m-’ SK’),

distributed uniformly over a square region around (- q, 0). The sides of this region have length q. The two-dimensional geometry is given in Fig. B2. The y-axis (see Fig. B2) is the boundary between wall and fluid.

The velocity profile of the saturated liquid is that of a turbulent flow in a channel with square cross section and a channel Reynolds number between 1 x lo4 and 6 x 104. An estimate for the time-averaged velocity at the cross section where the cavity is located is taken from the measurements of Melling and Whitelaw [lo]. From their experiments at Re = 4.2 x 10’ the following relation between axial velocity and position in the channel is derived :

Y

-;:. (-490) x Wdl Fluid

A

B

Fig. B2. Geometry used in calculation of liquid temperature distribution caused by a square heat source around (- q, 0)

and given liquid velocity distribution.

x 1/n

v=v, 7 0 (B3)

with I half the channel diameter, V, the liquid speed at the channel centre ; V, cz 1.25 V,,,, V,,,, being the liquid bulk speed. Coordinate x represents the distance from the wall (0 < x $ r). Melling and Whitelaw [lo] measured n = 5.36 for Re = 4.2 x 104. For other Reynolds numbers, a log- arithmic dependence of n on Re as in circular pipes (see for instance ref. [16]), is assumed :

n = 0,503 In (Re)

In is the natural logarithm. (W

The temperature boundary layer generated by the heat source embedded in the wall is calculated with boundary conditions 6 = 0 at boundaries A and B (see Fig. B2), while at C and D the normal heat flux is zero, i.e. %/an = 0, with n the normal to the wall. The solution is obtained with a finite element package with a mesh refined in regions with high temperature gradients. Upwind differencing is used to effectively handle the convective heat transport. It is found that the temperature boundary layer thickness only mar- ginally decreases with increasing liquid bulk speed at constant heating rate.

The difference in temperature of position (O,O), the bubble generation site, and the liquid bulk is calculated for eight mean liquid velocities between 0.05 and 1.0 m s-‘. The heating power, Q, of the bubble generator was set to the typical value of 10 W m-3. For the wall the material nronerties of glass were used (type I bubble generator), for 6e hquid the properties of water at saturation temperature at atmospheric pressure.

The calculated results for AT are very well represented by a power function :

AT = 0.6. Q * Vf,,,k (B5) with 5 = -0.568, Vbulk in m s-l, and Q in W rn3. This equation is equation (18). Because equation (30) is linear in 0, an increase in heating power leads to a proportional increase in AT, as has been made explicit by the factor Q in the above equation. Expression (BS) is used in equation (17) to give FAT.

forces on bubbles growing and detaching in flow along a ...it is found that either a vorticity lift...

Documents